Are there neural networks with very few nodes that decently solve non-trivial problems?
Even if it’s impossible to answer this question properly, as non trivial is not well defined (maybe the author will edit this questions later, to specify it better), I take the opportunity to point out this paper which looks interesting to me 

Smallest Neural Network to Learn the Ising Criticality

Assuming you have a general idea of the Ising Model I think the problem of identifying the critical temperature from a data driven perspective can be considered as non trivial and the paper shows how the authors have improved the performance related to solve this task with NN passing from 100 Hidden Neurons, as performed in this paper Machine learning phases of matter from 2017, to only 2 Hidden Neurons 

Just my cents: 


reducing the neurons, while keeping good performance, should help in terms of neural processing interpretability which is notoriously obscure and its complexity grows (exponentially) with the number of neurons